Ground-state properties and excitation spectra of non-Galilean-invariant interacting Bose systems

We study the ground state properties and the excitation spectrum of bosons which, in addition to a short-range repulsive two body potential, interact through the exchange of some dispersionless bosonic modes. The latter induces a time dependent (retarded) boson-boson interaction which is attractive in the static limit. Moreover the coupling with dispersionless modes introduces a reference frame for the moving boson system and hence breaks the Galilean invariance of this system. The ground state of such a system is depleted {\it linearly} in the boson density due to the zero point fluctuations driven by the retarded part of the interaction. Both quasiparticle (microscopic) and compressional (macroscopic) sound velocities of the system are studied. The microscopic sound velocity is calculated up the second order in the effective two body interaction in a perturbative treatment, similar to that of Beliaev for the dilute weakly interacting Bose gas. The hydrodynamic equations are used to obtain the macroscopic sound velocity. We show that these velocities are identical within our perturbative approach. We present analytical results for them in terms of two dimensional parameters -- an effective interaction strength and an adiabaticity parameter -- which characterize the system. We find that due the presence of several competing effects, which determine the speed of the sound of the system, three qualitatively different regimes can be in principle realized in the parameter space and discuss them on physical grounds.